A Sequence $a _ { 0 } , a _ { 1 } , a _ { 2 } , \ldots$

question 12

Essay

A sequence $a _ { 0 } , a _ { 1 } , a _ { 2 } , \ldots$ is defined recursively as follows:
$\begin{array} { l } a _ { 0 } = 2 , \quad a _ { 1 } = 9 \\a _ { k } = 5 a _ { k - 1 } - 6 a _ { k - 2 } \text { for all integers } k \geq 2\end{array}$
Use strong mathematical induction to prove that for all integers $$n \geq 0 , a _ { n } = 5 \cdot 3 ^ { n } - 3 \cdot 2 ^ { n }$$

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